Hmm sounds like a Webmail client, like Roundcube. Luckily (at least from my point of view) it has no 'unified inbox', but you can have as many mail accounts you want, with one login, from different vendors. You can selhost it easily. I use it on a Raspberry Pi with one login and have then access to gmail, yahoo and some other accounts.
To mimic a 'unified inbox' you can forward all the different accounts, to one 'major' account, so that you receive every mail in this inbox. Than you can create a 'sending alias', to answer the incoming mails with the proper SMTP service. Nothing easier than that with Roundcube.